I would be grateful if some member edit/edits my composition for me.
Which word in bold should I use? I'm undecided which word to use because I wonder whether it should be "...some member (could) edit..." (I mean 'could' is implied in the above sentence.)
I would be grateful if some member would edit my composition. If you want to leave out 'could/would', you could write: I would be grateful if some member edited my composition. This is subjunctive (since it hasn't happened yet); and the form is the same as the past tense form, as in 'If I were president...'
The subjunctive is not used as much in the form you are asking about; modal verbs are normally used instead.